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\includegraphics[max width=\textwidth, alt={}, center]{add3948c-3b45-4e67-ac84-e2ca935afd64-05_392_621_255_762} A particle \(P\) of mass 0.4 kg moves with constant speed in a horizontal circle on the smooth inner surface of a fixed hollow hemisphere with centre \(O\) and radius 0.5 m (see diagram).

1. Given that the speed of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its angular speed is \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\), calculate the angle between \(O P\) and the vertical.
2. Given instead that the magnitude of the force exerted on \(P\) by the hemisphere is 6 N , calculate
   (a) the angle between \(O P\) and the vertical,
   (b) the angular speed of \(P\).